How to extrude a region after I selected extrude vertices only?
When you extruded with Vertices Only there were created new vertices connected to existing vertices of the region:


Selected vertices moved away to show what actually is present after extruding

So you don't have any other option to extrude other than Vertices Only from here because technically there aren't any mesh elements selected except for vertices not forming neither faces nor edges. It's possible to extrude only another vertex from existing vertex (connected with an edge to something else or not).

To avoid creating wrong types of geometry delete these created vertices and select the face to extrude from it one more time. If you didn't move vertices after extruding there's an easy way to remove them, select all in that area and run W > Remove Doubles.